R-matrix propagation

Kulander, K.C. and Light, J.C. (1980). Photodissociation of triatomic molecules Application of the R-matrix propagation methods to the calculation of bound-free Franck-Condon factors, J. Chem. Phys. 73, 4337-4346. 

Baluja, K., Burke, P.G. and Morgan, L.A. (1982). R-matrix propagation program for solving coupled second-order differential equations, Comput. Phys. Commun. 27, 299-307. 

The coupled-channel equations (Eq. (31), (32), or (33)), may be solved in a variety of ways, but we use the R matrix propagation method of Light and Walker.Wo will review this method briefly in this section, as applied to the coupled channel equations in Jacobi coor dinates. The approach is essentially the same in other coordinates. 

Green s function (on O R A), the connection between this "local" R-matrix and the S-matrix must be made. For A in the as)nTiptotic region, this relation is given in Eq, (II.11), but we want A to be as small as possible for efficiency. We therefore consider below two methods of connecting the R-matrix to the S-matrix using a small "R-matrix box," i.e., small value of A. The first is a simple WKB (adiabatic) connection, and the second the more accurate R-matrix propagation [20,21]. 

The R-matrix propagation equations are obtained by assuming the R-matrix is known at Rj (in the basis of section i-1)... 

One apparent limitation of using the R-matrix propagation is that only (inverse) log derivative information is carried along—the wave-function is only determined asymptotically with the imposition of b.c. s. This makes the accumulation of overlap integrals for photodissociation, for example, somewhat more complicated. At this Workshop, however,... 

R. B. Walker [24] showed that the log derivative matrix can be propagated using the information evaluated in the R-matrix propagation, with a very similar algorithm. This permits the efficient propagation of overlap information in a relatively simple fashion analogous to that used by Kulander, et al. for collinear reactive photodissociation [25] and by Heather, et al. for 3-D triatomic photodissociation [26]. 

This gives a set of close-coupling equations coupled in the orbital quantum number Z and n. As before, we solve these equations using the R-matrix propagator method [22,26]. 

Much attention has been diverted to the numerical solution of the coupled-channel nations (3). Two of the most widely used and stable methods are the log-derivative approach introduced originally by Johnson[8] and the R-matrix propagator method developed by Light and Walker[9]. The log-derivative approach propagates the ratio between the derivative of the wavefunction (with respect to R) and the wavefonction, while the R matrix propagates the inverse of this quantity. Thus the log derivative matrix is defined by... 

R-matrix propagation method is a very stable technique, even when, many closed channels are necessary for convergence of the S matrix. 

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